Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and Dynamic Mode Decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

翻译：Koopman算子将非线性动力系统线性化，使其谱信息具有关键意义。人们已开发出大量算法来近似这些谱特性，而动态模式分解（DMD）作为投影方法的典型代表脱颖而出。尽管Koopman算子本身是线性的，但它作用于无穷维可观测函数空间这一事实带来了挑战，包括虚假模式、本质谱以及Koopman模式分解的验证问题。虽然近期研究已针对确定性系统解决了这些挑战，但在随机系统中仍存在显著空白——此类系统中Koopman算子度量的是可观测函数的期望值。我们证明，要解决这些问题必须超越期望。通过将方差纳入Koopman框架，我们应对了这些挑战。利用一个额外的DMD型矩阵，我们近似残差平方项与方差项之和，其中每一项均可通过批量快照数据独立近似。这使得随机Koopman算子谱特性的可验证计算成为可能，并实现了投影误差的控制。我们还引入方差-伪谱概念来评估统计一致性。最后，我们给出随机Koopman算子谱信息的一系列收敛性结果。本研究以模拟数据和实验数据的实际应用作结。在清醒小鼠的神经电生理记录中，我们展示了方差-伪谱如何揭示标准基于期望的动力学模型无法获取的生理学显著信息。